The Volturi are a group of ancient italian vampires that protect vampire law. In other words make sure the existence of vampires stay secret. The Volturi Coven includes Aro, Caius, Marcus, Sulpicia, and Athenodora. And Didyme used to but Aro killed her because he was too power hungry to let Marcus and her leave the Volturi. There's also The Volturi Guard. That consists of Alec, Jane, Felix, Demetri, Chelsea, Renata (partial). Basically they do all the REAL work. Aro, Marcus, and Caius aren't kick-ass enough to do much so they have others to do their bidding. In short Alec, Jane, Felix, and Demetri do pretty much everything and sometimes consult with Aro. In modern times (via Breaking Dawn) Aro has become tired of Plan A: Keep the existence of vampires a secret to Plan B: Take over the vampire world in power hungry in greedy ways.

Felix: Some n00bish vampires in England are creating an army
Aro: Perhaps we should take care of that.
Jane: Suppose we should to that.

*Jane, Alec, Demetri, and Felix go but a stop to (kill) the n00bish vampires meanwhile Aro, Marcus, Caius "leaders of the volturi" sit on their butts*